Infinite Oracle Queries in Type-2 Machines (Extended Abstract)

Infinite Oracle Queries in T yp e-2 Machines (Extended Abstract) Arno P auly ∗ Arno.Pauly@cl.cam.ac.uk No v em b er 10, 2021 W e define Oracle -T yp e- 2-Mac hine capable of writing infinite oracle queries. In con trast to fin ite oracle queries, this extends the realm of oracle-co mputable functions in to the discon tin uous r ealm. Our d efinition is conserv ativ e; access to a computable oracle d o es n ot increase the computational p ow er. Other mo dels of r eal hyp ercomputation su ch as Ziegler ’s (finitely) revis- ing computation or Typ e-2-No ndeterminism are shown to b e sp ecial cases of Oracle-T yp e-2-Mac hines. Our approac h offers an in tuitiv e definition of the w eak est mac hine mo del capable to sim ulate b oth T yp e-2-Mac hin es and BSS mac h ines. 1 Motivation & Overview As there are sev eral distinct notions of computabilit y for f u nctions on uncoun table sets (primarily T yp e-2-Mac hines [15] and BSS -machines [1]), a robust framew ork of hyper- computation could b e extremely u seful to allo w mutual comparisons. Ho we v er, so far a v ariet y of different concepts of real h yp ercomputation ha v e b een in tro duced and studied ([16], [18 ], [17], [6]). The desirable status of a p o werful unifi ed theory of real hyp ercom- putation has not b een reac hed ye t. The theory of discrete hypercompu tation is dominated b y the concept of orac le ma- c h ines, whic h give rise to the partial order of relativ e computabilit y on the T uring d egrees of problems. While a direct addition of d iscrete oracles to mac hines working on infin ite sequences w as u sed successfully to describ e the relatio nship betw een computabilit y and con tinuit y on infinite sequences, man y useful prop erties of oracle computation are lost in this process: Problems cannot b e c ompared using oracle mac hines, since p roblems and oracles are no longer are exc hangeable. Recen t progress on the study of W eihrauch degrees ([11], [3]) has unco v ered certain similarities to the theory o f T uring degree s; in f act t he order t heoretic prop erties of ∗ Computer Lab oratory , Universi ty of Cambridge, Cam bridge CB3 0FD, United Kingdom 1 W eihrauc h degrees are nicer, since they ev en form a (co mplete 1 ) distributiv e lattice . While it seems natural to consider W eihrauch degrees to represent degrees of incom- putabilit y , a corresp ond ing mo del of h yp ercomputation is missing. The ma jor obstacle f or obtaining s u c h a mo del is the f act that Oracle-T yp e-2-Mac hines w ould n eed the abilit y to p ose qu eries of infin ite length to the oracle and pro cess the (p ossibly infinite) answe r. Admitting compu tation steps to b e indexed by transfinite ordinals ω , ω + 1, ω + 2, . . . w ould allo w to con tin ue computation after the query has b een answ ered, ho w ev er, eve n without access to an oracle this would increase the computational p o w er 2 . In the p resen t paper w e will present a mac hine mo del that allo ws to pose infinite queries to oracles, without increasing th e computatio nal p o w er b eyond standard T yp e- 2-Mac hines in the case of co mputable oracles. W e will sho w ho w this mod el give s rise to W eihrauc h degrees as asso ciated r educibilit y . Sev eral previously in tro d uced concepts of real hyp ercomputation will b e shown to b e equiv alen t to oracle computatio n with resp ect to certain oracle s. 2 F oundations 2.1 The Mo del T o int ro duce the definition of an Oracle-T yp e-2-Mac hine, we start w ith recalling a formal definition of T yp e-2-M ac h ines. F or the sak e of sim p licit y , we consider the alphab et 3 P = { 0 , 1 } , one in put tap e, t w o working tap es and one output tap e. A Type-2-Mac h in e M is a lab elled directed graph, f u lfilling th e follo wing c onditions: 1. V er tices with out-deg ree t wo carry the lab els t i with i ∈ { 0 , 1 , 2 } . W e assume that the outgo ing e dges can be d istinguished, and will refer to the first or second successor. 2. V er tices with out-degree 1 c arry the lab els l i , r i with i ∈ { 0 , 1 , 2 } or w b i with i ∈ { 1 , 2 , 3 } and b ∈ { 0 , 1 } . 3. T here m u s t b e a un ique source with out-degree 1, lab elled s . 4. T he p ossible lab els f or sinks are a or r . There are n o v ertices w ith h igher out- degree. A configur ation of a T yp e-2-Ma c hine M is a tup le ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ), where q is a vertex in M , w i ∈ { 0 , 1 } N are infinite sequences, and n i ∈ N natural 1 While the conti nuous version of W eihrauch red ucibilit y allo ws the formation of arbitrary suprema and infima, for issues of uniformit y , the computable version only allo ws finite limits. 2 A machine computing LP O (s. Definition 23) could p roceed as follo ws: In St age 0, write a 0 in t he first cell of a w orking tap e. In stage n , chec k the n th input cell. If there is n ot a 0, write a 1 in th e first cell of the working tap e. In stage ω , copy the first cell of the w orking tap e to th e output tap e. Con tinue to output 0. 3 In some cases, the alphab et N is more conv en ien t. As standard encodings are av ailable, w e will neglect the details. 2 n umbers f or i ∈ { 0 , 1 , 2 , 3 , } . w i is to b e in terpreted as the cur ren t con ten t of the i th tap e, coun ted in th e order input tap e, first working tap e, second working tap e, output tap e; n i is the curren t p osition of the reading head. F or a v ertex q , w e let L ( q ) d enote its lab el, and S ( q ) its successor (or S 0 ( q ) its first and S 1 ( q ) its second su ccessor). F or a sequence w ∈ { 0 , 1 } N and a natural n um b er n ∈ N , w [ n ] denotes the n th symb ol in w . If additionally b ∈ { 0 , 1 } , then w \ [ n = b ] d enotes the sequence whic h is equ al to w in all p ositions except the n th, which is b . The standard relation → of single step transitions b et w een configur ations is d efined as follo ws: 1. ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) → ( S ( q ) , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ), if L ( q ) = s 2. ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) → ( S w i [ n i ] ( q ) , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ), if L ( q ) = t i 3. ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) → ( S ( q ) , w 0 , n 0 ± 1 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ), if L ( q ) = r 0 ( L ( q ) = l 0 and n 0 − 1 ∈ N ) 4. ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) → ( S ( q ) , w 0 , n 0 , w 1 , n 1 ± 1 , w 2 , n 2 , w 3 , n 3 ), if L ( q ) = r 1 ( L ( q ) = l 1 and n 1 − 1 ∈ N ) 5. ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) → ( S ( q ) , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 ± 1 , w 3 , n 3 ), if L ( q ) = r 2 ( L ( q ) = l 2 and n 2 − 1 ∈ N ) 6. ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) → ( S ( q ) , w 0 , n 0 , w 1 \ [ n 1 = b ] , n 1 + 1 , w 2 , n 2 , w 3 , n 3 ), if L ( q ) = w b 1 7. ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) → ( S ( q ) , w 0 , n 0 , w 1 , n 1 , w 2 \ [ n 2 = b ] , n 2 + 1 , w 3 , n 3 ), if L ( q ) = w b 2 8. ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) → ( S ( q ) , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 \ [ n 3 = b ] , n 3 + 1), if L ( q ) = w b 3 The relation ⇒ of arbitrary transitions is derived as the reflexiv e and transitiv e closure of → . No w there are t wo w a ys for a Type-2-Mac hin e to hav e a v alid output, b eing in a certain configuration. W e will d efi ne a generalized o utput relation dep ending on a transition relation, as th e details of our definition of the transitions are not relev ant for the definition of th e outp u t obtained f rom a certain configuration. F or finite computation, M yields the output w starting from a configuration ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) giv en a transition relation ⇒ , denoted as ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) ⇒ w , if there is a configuration ( ˆ q , ˆ w 0 , ˆ n 0 , ˆ w 1 , ˆ n 1 , ˆ w 2 , ˆ n 2 , w , ˆ n 3 ) w ith ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) ⇒ ( ˆ q , ˆ w 0 , ˆ n 0 , ˆ w 1 , ˆ n 1 , ˆ w 2 , ˆ n 2 , w , ˆ n 3 ) 3 and L ( ˆ q ) = a 4 . D ue to the definition, w and w 3 will b e equal exc ept for the p ositions b et w een n 3 and ˆ n 3 . F or infin ite computation, a configuration ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) yields out- put w given a transition relation ⇒ , denoted as ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) ⇒ w , if for eac h m ≥ n 3 there is a configuration ( q m , w m 0 , n m 0 , w m 1 , n m 1 , w m 2 , n m 2 , w m 3 , m ) with ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) ⇒ ( q m , w m 0 , n m 0 , w m 1 , n m 1 , w m 2 , n m 2 , w m 3 , m ) and lim m →∞ w m 3 = w . Note that the existe nce of suitable configurations alw a ys en sure the con v ergence of the sequence ( w m 3 ) m ∈ N . The defin ition of the output of a T u ring m ac hine M on input x is d efined as the output of M from configuration ( q , x, 0 , 0 N , 0 , 0 N , 0 , 0 N , 0) with L ( q ) = s . An Oracle-T yp e-2-Ma c h ine has a fur ther p ossible lab el ? for ve rtices with out-degree 2. Ho w ev er, th e d efinition of the transition relation and the result relation are n o w inter- t w ined. W e use → 0 to d enote the transition relation for n ormal T yp e-2-Mac hines, that is for → 0 v ertices lab elled ? effectiv ely ha v e the same effect has the r eje ct -v ertices lab elled r . W e no w define a sequence of transition relations and result relations ( → n , ⇒ n , ⇒ n ) n ∈ N inductiv ely . The oracle in u se is a multi- v alued f unction O : ⊆ { 0 , 1 } N ⇒ { 0 , 1 } N . → n alw ays includes → n − 1 , and add itionally the follo wing transitions: ( q , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , w 3 , n 3 ) → n ( S 0 ( q ) , w 0 , n 0 , w 1 , n 1 , y , 0 , w 3 , n 3 ) if ( S 1 ( q ) , w 0 , n 0 , w 1 , n 1 , w 2 , n 2 , 0 N , 0) ⇒ n − 1 x and y ∈ O ( x ) holds. ⇒ n is t he reflexiv e and transitiv e clo sure of → n . Informally , this means that a ?-v ertex sp a wn s a cop y of th e original mac h ine, with erased output tap e and with the s econd successor of the current verte x, r etriev es the result, feeds it to the oracle, and mo v es th e first successor v ertex, with the result of the oracle query b eing written on the second w orking tap e. F or eac h step of the inductiv e pro cess ab o ve, a multi -v alued fu n ction compu ted b y the T yp e-2-Oracle-Mac hine M with oracle O and query depth n can b e defin ed as: F n ( x ) = { y | ( q , x, 0 , 0 N , 0 , 0 N , 0 , 0 N , 0) ⇒ n y , L ( q ) = s } If O is single-v alued, then so will b e all F n . It is straigh t-forw ard to see that F n alw ays extends F n − 1 , th at is F n − 1 ( x ) ⊆ F n ( x ). If the sequence stabilizes, that is if there is an n 0 with F n 0 = F n for all n ≥ n 0 5 , w e call F n 0 the m ulti-v alued function computed by M and oracle O . 2.2 Basic p rop erties A fun damen tal requirement for our d efinition to matc h the intuitio n of oracle computa- tion is that an Oracle-T yp e-2-Mac hine with access to a computable oracle can compute 4 In most cases it is more con venien t to consider only the first ˆ n 3 symbols of w as the outp ut. In our case, how ever, this wo uld just make the defin ition of oracle calls more complicated. 5 Obviously , F n 0 = F n 0 +1 is already sufficient. 4 exactly those functions computable by a n ormal T yp e-2-Mac hine. As Typ e-2-Ma c hines are by our defi n ition a sp ecial case of O racle-T yp e-2-Ma c h ines (without ? b eing us ed as a label, all relatio ns → n are iden tical to → 0 ), w e certainly do n ot lo ose computational p o w er. How ev er, it migh t b e p ossib le that the m ec hanism used to p ose the oracle ques- tions could b e abused to p erform h yp ercomputation, ev en without a non-computable oracle. T o p ro v e t he con trary , w e mak e use of the n ext lemma. Inform ally , it states that, pro vided a fixed query depth, th ere is no need to use an y ve rtex in different qu ery la ye rs. Definition & Lemma 1 (Separation of Query Lay ers) . L et M b e an O racle-T yp e-2- Mac hine computing F with qu ery d epth n and oracle O . W e d efine another Or acle-T yp e- 2-Mac hine ˆ M computing F with query depth n and oracle O , in which eac h v ertex v is lab elled add itionally with a natural num b er N ( v ) ≤ n such th at the follo wing prop erties are fulfilled: 1. N ( S ( v )) = N ( v ) 2. N ( S 1 ( v )) = N ( v ) 3. N ( S 2 ( v )) = N ( v ) for v = t i 4. N ( S 2 ( v )) = N ( v ) + 1 for L ( v ) = ? W e make n id en tical copies of M without th e starting v ertex q with L ( q ) = s , and and n um b er all v ertices in the i th cop y with i . Starting with i = 1 an d p ro ceeding to i = n − 1 step by step, alw ays define S 2 ( v ) := S 2 ( w ) for all v ertices v with L ( v ) = ? and N ( v ) = i , where w is the corresp onding v ertex to v in the i + 1th cop y . F or all vertice s v with L ( v ) = ? and N ( v ) = n , replace the lab el ? b y r and delete the outgoing edges. The resulting graph forms an O racle-T yp e-2-M ac h ine ˆ M fulfilling the desired condi- tions. That ˆ M indeed c omputes the same function with query depth n as M does can b e c h ec ke d b y follo win g the definition of the result relation. Theorem 2. Let O be a computable oracle and M an Oracle-T yp e-2-Mac hine com- puting F w ith query depth n > 0. Then there is another Or acle-T yp e-2-Mac h ine M ′ computing F with oracle O and query depth n − 1. Pr o of. W e replace M by ˆ M due to Definition 1. W e consider all vertice s v with L ( v ) = ? and N ( v ) = n − 1. If there is none, then no v ertex can ha v e the lab el n , and we are done. Otherwise, we sho w h o w the num b er of su c h ve rtices can b e r educed b y one, iterated application yields the result. The construction corresp ond s to the one used for concatenation of Type-2-Mac hin es. Once the c hosen verte x v lab elled ? is reac h ed, a flag is set. Whenever a v ertex lab elled t 2 o ccurs, it will b e preceded by chec king th e flag: If it is not set, the normal v ertex is used. Otherwise, a certain T yp e-2-Mac hine is sim ulated 6 unt il it pro d uces the next 6 The t ap es needed for the sim ulated can b e encoded i nto the first w orking t ape without significant problems for the main compu tation. 5 output bit. The machine used corresp onds to the concatenation of the remaining mac hin e consisting of the v ertices lab elled b y n , with an additional in itial vertex conn ected to the former righ t successor of v , and the Typ e-2-Mac hine computing the o racle. Mov emen t commands on th e third tap e (wh ic h is the s econd wo rking tap e) are forwa rded to the second mac h ine, as well. F or this app roac h to w ork, only a fi nite num b er of sim ulations migh t b e concurren tly runn in g. F or multiple calls at the same la y er, eac h new call o verrides the result fr om the last call (sa ve the information that h as b een copied to the fir st w orking tap e), whic h allo ws to ab ort the asso ciated sim ulations. T o pr ev ent an infinite n umber of sim ulations o ccurring on d ifferen t nesting lev els, we had to restrict our considerations to finite query depth. Another desirable prop erty of an y defin ition of compu tational devices is closure un der comp osition. Here w e will sho w this prop ert y only f or those mac hin es p ossessing a distinguished query depth. F or those mac hines where eac h F n +1 is a prop er extension of F n , w eak er statemen ts can b e obtained in the same fashion. Theorem 3. Let M i b e an O racle-T yp e-2-Ma c h ine computing th e m ulti-v alued func- tion F i using the oracle O for i ∈ { 0 , 1 } . Then there is an Oracle-T yp e-2-Mac hin e M computing the m u lti-v alued function F 1 ◦ F 0 using the same oracle O . Pr o of. The stand ard pro cedure to concatenate T yp e-2-Mac hines can b e adapted directly to O racle-T yp e-2-M ac h ines. If the qu ery dep th n i is sufficien t to guaran tee stabilizatio n for the mac hine M i , then M stabilize s at n 0 + n 1 , as can b e verified directly follo wing the definitions. 2.3 Limiting the numb er of oracle queries While the query dep th can b e used to limit th e p o w er of an Oracle-T yp e-2-Mac hine, a finer distinction will prov e to b e usefu l. Instead of ju st limiting the nesting depth of queries, their n umb er can b e r estricted. Tw o d ifferen t concepts can b e pu rsued, e ither the total n um b er is c onsidered, or just the n u m b er of ca lls made at the top level of nestings. W e will primarily consider a limited n umber of oracle calls in cases where the query depth is limited to 1 anyw a y , in which case b oth notions coincide. 3 Relations of Relative Com putabi lit y Similar to the s ev eral differen t notions of r elativ e compu tabilit y used for T u ring m ac hin es and the corresp onding oracle mac h ines, one can in tro duce sev eral notions of r elativ e com- putabilit y us ing Oracle-T y p e-2-Mac hines. It turns out that s ome of the most int eresting ones coincide with r educibilities already suggested and stu died elsewhere. 6 3.1 Single Oracle Calls If a state lab elled ? ma y o ccur at m ost once dur ing a ny run of the Or acle-T yp e-2- Mac hine, w e can split the Oracle -T yp e-2-Mac hine into t w o regular Type-2-Mac h ines using Lemma 1: One computes th e oracle qu ery giv en the input, the other one compu tes the output, giv en the input and the answer t o the qu ery . Th u s, the corresp onding oracle computabilit y reduction coincides with computable W eihr auc h reducibility 7 ( ≤ W ), defined as: Definition 4. F or functions f , g ; f ≤ W g holds, if there are compu table partial f unctions F , G w ith f ( w ) = F ( w, g ( G ( w ))) ( w ∈ dom( f )). F or m ulti-v alued functions A , B ; A ≤ W B holds, if there are computable partial fu nctions F , G with x 7→ F ( x, g ( G ( x ))) ∈ A for all g ∈ B . Theorem 5. A (multi-v alued) function f is computable with a single c all to an oracle for g , if a nd only if f ≤ W g holds. The con tin uous v arian t ≤ W can b e obtained b y gran ting add itional access to an arbi- trary finite qu ery oracle, as this suffices to compute all con tinuous functions. This shows that the ≤ W -degrees of discon tinuit y can all b e r epresen ted b y a certain infin ite oracle, and vice ve rsa. Th us, all results known ab out ≤ W apply to infinite oracle computation (e.g. [7], [8], [13], [5], [9], [11], [14], [2], [3], [4], just to list some of them). One feat ure that s h all be p ointed out is tr ansitivit y: A ≤ W B and B ≤ W C imp lies A ≤ W C . Th us sets lik e Al l (multi-v alue d) functions c omputable using a single query to the or acle O are interv als r egarding ≤ W . 3.2 Finitely many (indep endent) o racle calls with query depth 1 If one considers more than one call to the oracle, the query d epth b ecomes imp ortan t again. F or the sake of simplicit y , w e will alw a ys assume the query depth to be fi x ed to 1 in the follo wing. Already for t w o allo wed oracle calls things get more complicated: One part of the Oracle-T yp e-2-Mac hine computes the query f or the fir s t call fr om the input, the sec- ond part compu tes the sec ond query from th e input and the result of the first call, the third part tak es all information a v ailable and pro du ces the output. Th u s the cor- resp ondin g reducibilit y w ould ask for three computable fun ctions F , G , H , so that f ( x ) = F ( x, g ( G ( x )) , g ( H ( x, g ( G ( x ))))). In addition, the relation f is c omputable with at most n or acle c al ls to g is not transitiv e an ymore, d iminishing its app eal for fu r ther consideration. Instead, w e will consider three v ersions of relativ e computabilit y whic h eac h con tain a restriction to an unsp ecified but finite n um b er of oracle calls. Pro vided that the d ifferent oracle calls to n ot d ep end on eac h o ther, the cartesian pro du ct of (m ulti-v alued) fu nctions can b e emplo y ed. The follo wing theorem will p r o vid e 7 In previous w ork, this reducibility has also b een called W adge reducibilit y ( ≤ w ) or T yp e-2-Reducibility ( ≤ 2 ). 7 an imp ortan t result enabling the use of cartesian pro ducts to gether with W eihrauc h reducibilit y . Definition 6. F or fu nctions f : X 1 → Y 1 , g : X 2 → Y 2 , define h f , g i : ( X 1 × X 2 ) → ( Y 1 , Y 2 ) b y h f , g i ( x 1 , x 2 ) = ( f ( x 1 ) , g ( x 2 )). F or multi-v alued functions F , G , d efine h F , G i as the s et {h f , g i | f ∈ F , g ∈ G } . De fine h f i n and h F i n b y iteratio n. Theorem 7. f ≤ W g implies h f i n ≤ W h g i n for eac h n ∈ N . Pr o of. If f ( x ) = F ( x, g ( G ( x ))) holds, th en also h f i n ( ¯ x ) = h F i n ( ¯ x, h g i n ( h G i n ( ¯ x ))) is true. As computabilit y of a function F implies compu tabilit y o f h F i n , th is completes the pro of. Instead of intro d ucing a formal d efi nition of ind ep endent oracle calls, w e w ill consid er a class of oracles for whic h indep endence is n ot necessary to arriv e at a succinct notion of relativ e c omputabilit y . If the oracle has only a fi nite n um b er of possib le answers, then indep enden ce can b e obtained by an exp onen tial in crease in the n umber of queries: Replace th e second query by several queries, one for eac h p ossible answe r to the first query , and so on. There are three different definitions o f r elatively c omputable using only finitely many or acle c al ls . The num b er of oracle calls could b e b ounded by a constant ind ep endent of the actual inp ut, they could b e b oun ded b y a compu table fun ctions defined on the input, or un b ound ed , but guaran teed to b e finite. The fir s t relation is: Definition 8. Let f ≤ bc W g holds, if there is an n ∈ N , so th at f ≤ W h g i n holds. A ≤ bc W B holds, if there is an n ∈ N , so that A ≤ W h B i n holds. Theorem 9. A (m u lti-v alued) function f is computable with a fixed finite num b er of oracle calls to g with | range( g ) | < N , if an d only if f ≤ bc W g holds. Theorem 10. ≤ bc W is transitiv e. Pr o of. W e will pro v e the claim just for functions, the p ro of for m ulti-v alued fu nctions pro ceed an alogously . Assume f ≤ W h g i n and g ≤ W h h i m . Applicatio n of Theorem 7 yields h g i n ≤ W hh h i m i n . T rivial consideration is enough to see hh h i m i n ≡ W h h i nm , th us we h a ve h g i n ≤ W h h i nm . T ransitivit y of ≤ W is used to obtain f ≤ W h h i nm , which implies f ≤ bc W h . The relation of relativ e computabilit y where th e num b er of oracle calls is b oun ded b y a computable f unction defined on th e input wa s su ggested in [11, Sub section 6.1] as ≤ ct , f or th e sake of consistency w e will call it ≤ bf W here. F or defin in g it w e w ill need the supremum for ≤ W , whic h coincides with the copro d uct of fun ctions: 8 Definition 11. Let ( f i ) i ∈ N b e a co unta ble family of functions. Define ⌈ f i ⌉ i ∈ N through ⌈ f i ⌉ i ∈ N ( ix ) = if ( x ). F or a coun table f amily ( F i ) i ∈ N of m ulti-v alued functions, defin e ⌈ F i ⌉ i ∈ N through: ⌈ F i ⌉ i ∈ N = {⌈ f i ⌉ i ∈ N | ∀ i ∈ N f i ∈ F i } Definition 12. Let f ≤ bf W g hold, if f ≤ W ⌈h g i n ⌉ n ∈ N holds. Let A ≤ bf W B h old, if A ≤ W ⌈h B i n ⌉ n ∈ N holds. Theorem 13. A (m ulti-v alued) fun ction f is computable with a fin ite n um b er of oracle calls b ound ed by a computable fu nction to g with | range( g ) | < N , if and only if f ≤ bf W g holds. Theorem 14. ≤ bf W is transitiv e. Pr o of. Again, the pro of w ill be done only f or f unctions, for m ulti-v alued fun ctions one can proceed analogo usly . Assume f ≤ bf W g and g ≤ bf W h . By definition, this means f ≤ W ⌈h g i n ⌉ n ∈ N and g ≤ W ⌈h h i m ⌉ m ∈ N . Ob serv e the distrib u tivit y la w [11, T heorem 6.2]: h f , ⌈ g i ⌉ i ∈ N i ≡ 2 ⌈h f , g i i⌉ i ∈ I T ogether with Theorem 7 w e thus h a ve: h g i n ≤ W h⌈h h i m ⌉ m ∈ N i n ≡ 2 ⌈h h i m ⌉ m ∈ N As this h olds for all n ∈ N , the p r op ert y of ⌈ ⌉ b eing the supremum in the partial ord er ≤ W yields ⌈h g i n ⌉ n ∈ N ≤ W ⌈h h i m ⌉ m ∈ N . T ransitivit y of ≤ W no w completes the pr o of. Also the third v ersion of r elatively c omputable with finitely many or acle c al ls c an b e expressed u sing ≤ W and a certain construction deriv ed from the parallelizatio n intro- duced in [3, Section 4]. W e will s tart with defining the parallelizatio n of a (m ulti-v alued) function defined on N N . F or that, we fix a homeomorphism λ : ( N N ) N → N N . Definition 15. Giv en a m ulti-v alued fun ction F : ⊆ N N → N N , d efine ¯ F : ⊆ ( N N ) N → ( N N ) N through ¯ F ( Q n ∈ N x n ) = Q n ∈ N F ( x n ). Then defin e ˆ F : ⊆ N N → N N via ˆ F = λ ◦ ¯ F ◦ λ − 1 . The v arian t we need is obtained by prerestricting λ to the set { w ∈ ( N N ) N | |{ i ∈ N | w ( i ) 6 = 0 N }| < ∞} . Th e restriction s h all b e denoted λ < ∞ . Then we can contin ue to define: Definition 16. Given a m ulti-v alued f unction F : ⊆ N N → N N , defi ne ˆ F < ∞ : ⊆ N N → N N via ˆ F < ∞ = λ ◦ ¯ F ◦ ( λ < ∞ ) − 1 . T o prepare for the p ro of of the transitivit y of the corresp ondin g reducibilit y relation, w e sh o w th at ˆ . < ∞ is a closure op erator regarding ≤ W , similar to [3, Prop osition 4.2]. The statemen ts hold b oth for functions and m ulti-v alued functions. Theorem 17. 1. f ≤ W ˆ f < ∞ 9 2. f ≤ W g implies ˆ f < ∞ ≤ W ˆ g < ∞ . 3. d ˆ f < ∞ < ∞ ≤ W ˆ f < ∞ Pr o of. The pro of is exactl y analogous to the pro of of [3, Prop osition 4.2]. Definition 18. Let f ≤ f W g h old, if f ≤ W ˆ g < ∞ holds. Let A ≤ f W B hold, if A ≤ W ˆ B < ∞ holds. Theorem 19. A (m ulti-v alued) function f is computable w ith an y finite num b er of oracle calls to g with | range( g ) | < N , if an d only if f ≤ f W g holds. Theorem 20. ≤ f W is transitiv e. Pr o of. Assume A ≤ f W B and B ≤ f W C . Then we ha v e B ≤ W ˆ C < ∞ . App licatio n of Theorem 17 2. and 3. yields ˆ B < ∞ ≤ W ˆ C < ∞ , b y trans itivit y of ≤ W one can obtain A ≤ W ˆ C < ∞ , w hic h by definition is A ≤ f W C . 3.3 Infinitely many oracle calls with fixed query depth Infinitely man y oracle calls to a function w ith fi nite r ange and query depth 1 yields the relation ≤ ˆ W from [3, Defin ition 4.3]. F or its prop er ties, w e refer to [3]. Definition 21. Let f ≤ ˆ W g hold, if f ≤ W ˆ g holds. Let A ≤ ˆ W B hold, if A ≤ W ˆ B holds. Theorem 22. A (m ulti-v alued) function f is computable with in finitely m an y oracle calls with nesting d epth 1 to g with | range( g ) | < N , if and only if f ≤ ˆ W g h olds. 4 Using LP O as o racle The omniscience principle LP O and its equiv alence class h av e receiv ed a lot of atten tion in th e literature, partly motiv ated b y the fact that LP O is th e least discon tin u ous function defined on a sep arable space. In this section, we will explore the p ow er of oracle access to LP O , applying the differen t restrictions in tro duced so far. Definition 23. Define LP O : { 0 , 1 } N → { 0 , 1 } N b y LP O (0 N ) = 0 N and LP O ( w ) = 10 N for w 6 = 0 N . As t he range of LP O is finite, t he relati ons for multiple oracle queries introd uced in Section 3 can b e used here. 10 4.1 Classical o racle computation As w e can us e a Typ e-2-Ma c hine to compute fun ctions f : N → N in exactly the same w a y as classical T uring m ac hines, it is an in teresting question w hic h p o w er access to uncounta ble oracles suc h as LP O pro v id es. It is easy to see that the halting problem ∅ ′ can b e solve d with a single q u ery to LP O : Start the oracle call. Simulate the mac hine giv en as inp ut, printing 0 f or eac h step it do es not halt. If it halts, print 1, and con tin u e to prin t 0s. Then the oracle returns 0 N , if the mac h ine halts, and 10 N otherwise. No w assu me a function f : N → N with f ≤ W LP O . In the pro cess of w r iting the oracle call, an oracle mac hine computes a function G : N → { 0 , 1 } N . Th is fu n ction G can b e mo dified to yield a computable partia l function G ′ : ⊆ N → N , that h alts if and only if G writes a 1. T h us, the set G − 1 (0 N ) ⊆ N is co-recursiv ely en umerable. This implies the existence of a computable fu nction H w h ic h n ∈ G − 1 (0 N ) if and only if H ( n ) / ∈ ∅ ′ . This sho ws that f can b e computed by an oracle mac hin e that mak es o ne call to ∅ ′ . As all functions computable with oracl e a ccess to ∅ ′ are still c on tin uous, LP O is n ot computable w.r.t. ∅ ′ . Therefore, at least for the sp ecial ca se of exactl y one oracle call to L P O , we arrive d at an extension of the classical degrees of oracle computabilit y that, restricted to the classica l case, coincides with the original definition. 4.2 Fixed finite numb er of queries to LP O The functions h LP O i n that arise here are iden tical to the functions LP O n in tro duced in [14]. As demonstrated in [5], LP O n is complete f or the set of functions with Lev el less or equal than n + 1. W e will consider the relationship b et ween finitely many oracle calls to LP O and the Level of a function in further detail. If the num b er of oracle queries the mac hine can mak e is fixed in adv ance to n , then w e can sp lit the Oracle-T y p e-2-Mac hine with n queries into an Oracle-T yp e-2-M ac h ine with n − 1 quer ies and a Typ e-2-Ma c hine, the latter computing the first oracle qu er y from the in put, the form er computing the outpu t giv en the inpu t and the answer to the first query . In the case of LP O b eing the oracle, th is corresp onds to th e Ω n -con tinuous functions studied in [7]. The pro cess describ ed ab o v e to replace oracle calls by indep enden t oracle calls can b e used to der ive an exp onen tial u pp er b ound for the level of a function. Ho w ev er, w e will used th e decomp osition describ ed in the last paragraph, recycling a related proof from [9]. Theorem 24. If f can b e computed by an O racle-T yp e-2-Ma c h ine making not more than n call s to an oracle for LP O , then Lev ( f ) ≤ 2 n . Pr o of. In the case n = 1 w e hav e f ≤ 2 LP O , together with results from [5] the claim follo ws. F or the ind uction step, assume that th e (m u lti-v alued) fun ction computed b y the machine using n − 1 queries is F . W e ha v e L 2 i ( f ) ⊆ { x | ( x, 0 N ) ∈ L i ( F ) ∨ ( x, 10 N ) ∈ L i +1 ( F ) } a nd L 2 i +1 ( f ) ⊆ { x | ( x, 10 N ) ∈ L i ( F ) ∨ ( x, 0 N ) ∈ L i +1 ( F ) } , yieldin g Lev( f ) = 2 Lev ( F ) = 22 n − 1 = 2 n . 11 4.3 Bounded finite n u mb er of queries to LP O Making a n u m b er of oracle calls to LP O that is b ound ed b y a computable fu nction is, as exp lained ab o v e, equ iv alen t to a single o racle ca ll to ⌈h LP O i n ⌉ n ∈ N . A more natural complete p roblem for this class is fin ding the minimal num b er in an unsorted infinite sequence of n atural n um b ers, see [9]. 4.4 Any finite numb er o f queries If the Or acle-T yp e-2-Mac h ine can make an y finite num b er of oracle queries to LP O , one deriv es an mo del equiv alen t to finitely revising computation pr esen ted in [18]. The same functions are computable b y a sin gle oracle qu ery to Max . Definition 25. Define the p artial fu nction M AX : ⊆ N N → N through M AX ( w ) = max { w ( i ) | i ∈ N } . Theorem 26. A single oracle query to Max is reducible to an y fin ite n umber of oracle queries to LP O . Pr o of. F or an y natural n umber n and inp ut sequence w , let w n b e th e sequence d efined b y w n ( i ) = ( 0 w ( i ) ≤ n 1 else . A mac h ine calls the oracle LP O on w n for eac h n , until the first cell of th e oracle answer cont ains 0 for the first time. Then n is the correct outpu t for Max . As long as w was a v alid input for Max , this happ ens in a finite num b er of steps. Theorem 27. Finitely r evising computation can sim ulate an y finite num b er of oracle calls to LP O . Pr o of. Start with sim ulating the Oracle-T yp e-2-Mac h ine. Whenev er an o racle c all is encoun tered, contin ue to simulat e the main computational thread of the oracle mac h ine assuming that the oracle answ ered 0 N . In parall el, compute the oracle qu ery . If during the computation of any of the finitely many oracle queries another symbol than 0 results, ab ort the output written so far, return to the moment in wh ich the computation of the resp ectiv e qu ery wa s started, and restart from th er e, using 10 N as the ans wer from the oracle no w. Theorem 28 . Finitely revising computation can b e simulated b y a single oracle call to Max . Pr o of. W e only need to sh ow that the trans lation from ˆ ι to ι can b e computed by suc h an oracle mac hine. Compu te the oracle q u ery by reading the in p ut and pr in ting the highest index of a revising mark found sofar. Once the answ er n is obtai ned from the oracle, discard the first n symb ols from the input and output the rest. 12 This sho ws that the degree of discont in uit y of revising computation is the least dis- con tinuous bu t discont in uous one that is closed und er composition of fun ctions. Since the comp osition of functions computable b y BSS machines ([1]) is compu table by a BSS mac h ine, th e corr esp onding degree of discon tin uit y must con tain the one considered here. The other inclusion holds as we ll, replicating a result from [17]: Theorem 29. An Or acle-T yp e-2-Mac h ine making a finite n um b er of oracl e calls to LP O can sim ulate a BSS m ac hine. Pr o of. An ordin ary T yp e-2-Mac hine can sim ulate every computation step of a BSS ma- c h ine except tests on equalit y . T esting t w o real num b ers for equ ality is equiv alen t to LP O , so a corresp ondin g oracle call allo ws an Oracle-T yp e-2-Mac hine to sim ulate all steps of a BS S mac hine. If one searches f or a mo del of computation incorp orating b oth the capabilities of BSS mac h ines and Typ e-2-Mac hines, without introdu cing u nnecessary additional p o wer, one arriv es at an Oracle-T yp e-2-Mac hine making a finite n umber of oracle calls to LP O , making this a very p r omising mac hine mo del for the study of algorithms on the rea l n umbers. Theorem 30. The set of fun ctions computable w ith finitely m any oracle calls to LP O is the smallest set closed un der comp osition and pro ducts con taining the T yp e-2-computable and the BSS -computable functions. Pr o of. Due to Th eorems 26, 27, 28 the said set of fu n ctions is the set of functions computable with a single oracl e call to Max . Due to Theorem 5, an y function in this set is of th e f orm x 7→ F ( x, M AX ( G ( x ))) , where F and G are computable b y a T yp e-2- Mac hine. As th e function M AX is computable by a BSS-mac hine, the set is minimal. 4.5 Infinitely many oracle queries to LP O The parallelization [ LP O w as stu died as C in [13]. There are a wid e v ariet y of p roblems that turned out to b e equiv alent to C , we refer to [3] for a con temp orary o v erview. An equiv alen t mo del of h yp er-computation is the α ′ -computabilit y from [16], [18]. A higher n esting d epth, fixed to n , corresp ond s to th e s tand ard generalizatio ns of the notions discussed ab o ve: A complete function is C n (obtained as n -times the co ncate- nation of C ) and the mo d el of hyper-compu tation is α n -computabilit y . As sh o wn in [2], the corresp onding degree of discon tinuit y is the set of P n -measurable functions. 5 Other Mo dels o f Hyp ercomputation So far we hav e discuss ed ho w (finitely) revising computation can b e expressed as u s e of oracle calls to LP O . Other models of hyp ercomputation are expressible in our frame- w ork, as well. W e start with Type-2-Nondeterminism as in tro duced by Ziegler . 13 Definition 31. Th e problem UnProject tak es a name of a T yp e-2-Mac hine M and an infinite s equence x , and asks for an infinite sequence y , so that M accepts h x, y i . Theorem 32. A single oracle call to UnProject is equiv alen t to T yp e-2-Nondeterminism. Pr o of. UnProject can easily b e solv ed b y a nondeterministic T yp e-2-Mac hine: Guess y , sim ulate M on input h x, y i . If M rejects, ab ort the computation. If M accepts, cop y y on the output tap e. F or th e other direction, the nond eterministic Type-2-Mac h ine can b e split in t w o parts: The fi rst part verifies the guess, and is used as input for U nPr oject toge ther with the actual in put string. The second part uses the guessed sequence (or, alternativ ely , the output of UnProject ) and the input to compute the outpu t. Theorem 33. \ UnProject < ∞ ≡ W UnProject . Pr o of. One d irection is trivial. F or the other direction, giv en a finite collection of n T yp e-2-Mac hines a Type-2-Mac h ine with n input tap es can b e constructed that accepts, if the i th Typ e-2-Ma c hine accepts the in put on the i th inp u t tape for all i ≤ n . T his mac h ine is used as input for UnProject together with th e pro du ct of th e n infinite sequences. Whether UnProject is eve n equiv alent to \ UnProject is left op en. An answ er to this question w ould shed a ligh t on the robustness of T yp e-2-Nondeterminism. The r esults obtained so f ar defin itely show that T y p e-2-Nondeterminism can b e equiv- alen tly expressed in dete rministic terms. F ollo wing the parallels drawn by Ziegler b et w een nondeterministic Buec h i-automata and nondeterministic T yp e-2-Mac hines in claiming that nond eterminism might b e the more app ropriate choic e f or infinite com- putation, we r efer to deterministic parit y automata 8 and deterministic Oracle-T yp e-2- Mac hines with fi nite oracle access to UnProject ; p oin ting out that nond eterminism can b e av oided in b oth cases. An example for a problem not solv able b y any analytic al mac hine is the stabilit y of a dynamical system ([6]). T he task of co nstructing an Or acle-T yp e-2-Mac h in e capable of s olving it is trivial: J u st ad m it a single oracle call to the problem itself. Pote nt ial further researc h w ould consider which problems are reducible to it, whether more access to the same oracle increases the computational p ow er, and so on. 6 Applications 6.1 Arithmetic Circuits An application for Oracle-T yp e-2-Mac hin es outside the usual realm of the T yp e-2-Theory of Comp u tabilit y are giv en b y Arithmetic Circuits as defined in [12]. Instead of the 8 A language is expressible by a n on d eterministic Buechi automaton, if and only if it can b e expressed by a parity automaton. 14 usual gates used in Ar ithmetic Circuits, we will r ep lace the m ultiplication gate by the t wo follo wing: A contin uous m ultiplication gate: A × B := { n ∗ m | n ∈ A m ∈ B } ∪ { 0 } and a (discon tinuous) test gate: T ( A ) = ( ∅ A = ∅ { 0 } otherwise Usual m u ltiplication can b e expressed b y th e tw o new gates, an d b oth of the n ew gat es can b e expr essed by the stand ard gates used in Arithm etic Circu its, so our mo dified circuits can define exactly the fu nctions n ormal Arith m etic Circuits can define. An y Arithmetic C ir cuit u s ing n test ga tes can obviously be sim ulated b y an Or acle- T yp e-2-Mac hine making n calls to an oracle for LP O . An application of Theorem 24 yields the fac t that functions d efinable b y Arithmetic Circuits alw ays ha v e finite lev el. This result directly implies many of the resu lts presen ted in [12], others f ollo w from th e observ ation that the lo ok-ahead 9 needed to simulate an Arithmetic Circuit is b ound ed b y l ( n ) = n . 6.2 The degree of discontinuit y of Nash equilibr ia While the language needed to state results regarding the d egree of incomputabilit y or discon tin uit y of problem was presen t for almost t w o decades in the form o f W eihrauch- reducibilit y , the concept of Oracle-T yp e-2-Ma c hines allo ws new pro of st yles suitable to arriv e at new results. An example of suc h w ork is [10], where O racle-T yp e-2-M ac h ines are used to study the d egree of discon tin uit y shared by m ultiple robust divisions, solving systems of linear inequalities and finding Nash and correlated equilibria in (zerosum) games. References [1] Lenore Blum, Mik e Sh ub, and Stev e Smale. 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